(2+1)-dimensional gravity with the cosmological constant (量子重力(研究会報告))

概要

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Following the ADM canonical formalism, we analyzed the (2+1)-dimensional gravity with the cosmological constant, where the 2-dimensional space is a compact Riemannian manifold M without a boundary. York's time slice γ=K=constant on the 2-dimensional spatial manifold M was adopted. We see that in this time slice solving the Hamiltonian and momentum constraints is equivalent to considering a conformally equivalent class of superspace. Thus we find that the global dynamical mode is Teichmuller deformation of M and it is directly related to the topology of space. Then using some mathematical facts on the Teichmuller space of a closed Riemannian manifold, we have reduced the classical dynamics of gravity to that of a poin-particle theory and explicitly followed the time evolution equations in the case-M is topologically a torus. Our main findings concerning the Teichmuller deformation are the two facts. For torus, we have explicitly shown that (1) the trajectory of the Teichmuller parameters is a geodesic of the Teichmuller space even in the presence of the cosmological constant (2) the cosmological constant causes a convergence effect of the geodesic motion. That is, the Teichmuller deformation stops asymptotically in time. However, we do not know yet at present how to describe higher genus cases and if there are any possibility of spatial topology change, a more challenging problem.
素粒子論グループ素粒子研究編集部の論文
1990-09-20